Group #13144201 — subgroups

Subgroups — #13144201

Order 1728.

462 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#11721776	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88386	C₂	1	1	yes	maximal, characteristic, complemented
4	#11720511	C₂	1	2	yes	maximal, complemented
5	C₁₆×#272	C₂	1	2	yes	maximal, complemented
6	C₃×#4907	—	3	4	no	maximal
7	#88740	C₄	1	1	yes	characteristic
8	C₂×C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	#88538	C₄	1	1	yes	characteristic
10	#88527	C₄	1	2	yes	—
11	C₈×#272	C₂²	1	2	yes	—
12	C₁₆×H(3,𝔽₃)	C₂²	1	2	yes	—
13	#88386	C₂²	1	1	yes	characteristic
14	#88386	C₂²	1	1	yes	characteristic
15	C₃×#901	—	3	4	no	—
16	C₃²×M₃₂	D₃	1	4	yes	complemented
17	C₂×C₃×C₃⋊C₁₆	—	3	4	no	—
18	C₃×#798	—	3	8	no	—
19	C₃×D₃×C₁₆	—	3	8	no	—
20	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
21	#6235	C₈	1	2	yes	—
22	C₄⋊#272	C₈	1	1	yes	characteristic
23	C₄×#272	C₂×C₄	1	2	yes	—
24	#6229	C₈	1	1	yes	characteristic
25	C₈×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
26	C₈×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
27	#6180	C₂×C₄	1	2	yes	—
28	C₃×#504	—	9	1	no	—
29	#4907	—	3	4	no	—
30	C₃×#235	—	3	4	no	—
31	C₂×C₃²×C₈	C₂×D₃	1	4	yes	—
32	C₃×#208	—	3	4	no	—
33	C₃×ΓL(1,𝔽₂₅)	—	3	8	no	—
34	C₃×D₃×C₈	—	3	8	no	—
35	C₃²×C₁₆	C₂×D₃	1	8	yes	—
36	C₃×C₃⋊C₁₆	—	3	4	no	—
37	C₃×C₃⋊C₁₆	—	3	4	no	—
38	C₂²×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
39	C₂×#272	C₂×C₈	1	2	yes	—
40	C₄×H(3,𝔽₃)	C₂×C₈	1	1	yes	characteristic
41	C₄×H(3,𝔽₃)	C₂²×C₄	1	1	yes	characteristic
42	#1084	C₂×C₈	1	2	yes	—
43	C₃×#131	—	9	1	no	—
44	#901	—	3	4	no	—
45	C₃×M₃₂	—	3	4	no	—
46	C₃×M₃₂	—	9	2	no	—
47	C₃×M₃₂	C₃⋊D₃	1	1	yes	characteristic
48	C₂×C₃×C₁₆	—	9	1	no	—
49	C₂×C₃×C₁₆	—	9	2	no	—
50	C₂×C₃⋊C₁₆	—	3	4	no	—
51	#798	—	3	8	no	—
52	D₃×C₁₆	—	3	8	no	—
53	C₂×C₃²×C₄	C₄×D₃	1	4	yes	—
54	C₃×#67	—	3	8	no	—
55	C₃×D₁₂	—	3	4	no	—
56	C₃×C₄×D₃	—	3	8	no	—
57	C₃×Dic₆	—	3	4	no	—
58	C₃²×C₈	C₄×D₃	1	4	yes	—
59	C₃²×C₈	C₂²×D₃	1	4	yes	—
60	C₃×C₃⋊C₈	—	3	8	no	—
61	#504	—	9	1	no	Sylow
62	C₂×H(3,𝔽₃)	—	2	1	no	—
63	C₂×H(3,𝔽₃)	C₂²×C₈	1	1	yes	characteristic
64	#272	—	2	2	no	—
65	C₃×#41	—	9	1	no	—
66	#235	—	3	4	no	—
67	C₃×M₁₆	—	9	1	no	—
68	C₃×M₁₆	—	9	2	no	—
69	C₂×C₃×C₈	—	3	4	no	—
70	C₂×C₃×C₈	—	9	2	no	—
71	C₂×C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
72	#208	—	3	4	no	—
73	ΓL(1,𝔽₂₅)	—	3	8	no	—
74	D₃×C₈	—	3	8	no	—
75	C₃×C₁₆	—	3	8	no	—
76	C₃×C₁₆	—	9	1	no	—
77	C₃×C₁₆	—	9	1	no	—
78	C₃×C₁₆	C₂×C₃⋊D₃	1	2	yes	—
79	C₃⋊C₁₆	—	3	4	no	—
80	C₃⋊C₁₆	—	3	4	no	—
81	C₂²×C₃²	D₃×C₈	1	4	yes	—
82	C₂×C₃×D₃	—	3	8	no	—
83	C₃²×C₄	D₃×C₈	1	4	yes	—
84	C₃²×C₄	C₂×C₄×D₃	1	4	yes	—
85	C₃×Dic₃	—	3	8	no	—
86	#131	—	9	1	no	—
87	M₃₂	—	9	2	no	—
88	M₃₂	#272	1	1	yes	characteristic, complemented
89	C₂×C₁₆	—	9	1	no	—
90	C₂×C₁₆	—	9	2	no	—
91	H(3,𝔽₃)	#504	1	1	yes	characteristic, complemented, Sylow
92	C₃×Q₈	—	9	1	no	—
93	C₃×D₄	—	9	2	no	—
94	C₃×D₄	—	9	1	no	—
95	C₂×C₃×C₄	—	3	4	no	—
96	C₂×C₃×C₄	—	9	2	no	—
97	C₂×C₃×C₄	C₄×C₃⋊D₃	1	1	yes	characteristic
98	#67	—	3	8	no	—
99	D₁₂	—	3	4	no	—
100	C₄×D₃	—	3	8	no	—
101	Dic₆	—	3	4	no	—
102	C₃×C₈	—	3	4	no	—
103	C₃×C₈	—	3	4	no	—
104	C₃×C₈	—	9	2	no	—
105	C₃×C₈	C₄×C₃⋊D₃	1	1	yes	characteristic
106	C₃×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
107	C₃⋊C₈	—	3	8	no	—
108	C₂×C₃²	—	2	4	no	—
109	C₂×C₃²	C₂×D₃×C₈	1	4	yes	—
110	C₃×D₃	—	6	8	no	—
111	#41	—	9	1	no	—
112	M₁₆	—	9	1	no	—
113	M₁₆	—	9	2	no	—
114	C₂×C₈	—	9	2	no	—
115	C₂×C₈	C₂×#272	1	1	yes	characteristic
116	C₁₆	—	9	1	no	—
117	C₁₆	—	9	1	no	—
118	C₁₆	C₂×#272	1	2	yes	—
119	C₂²×C₃	—	3	4	no	—
120	C₂²×C₃	—	9	2	no	—
121	C₂²×C₃	C₈×C₃⋊D₃	1	1	yes	characteristic
122	C₂×D₃	—	3	8	no	—
123	C₃×C₄	—	3	4	no	—
124	C₃×C₄	—	3	4	no	—
125	C₃×C₄	—	9	2	no	—
126	C₃×C₄	C₈×C₃⋊D₃	1	1	yes	characteristic
127	C₃×C₄	C₂×C₄×C₃⋊D₃	1	1	yes	characteristic
128	Dic₃	—	3	8	no	—
129	C₃²	#4907	1	4	yes	complemented
130	Q₈	—	9	1	no	—
131	D₄	—	9	2	no	—
132	D₄	—	9	1	no	—
133	C₂×C₄	—	9	2	no	—
134	C₂×C₄	C₄×#272	1	1	yes	characteristic
135	C₈	—	9	2	no	—
136	C₈	C₄×#272	1	1	yes	characteristic
137	C₈	C₂²×#272	1	1	yes	characteristic
138	C₂×C₃	—	2	1	no	—
139	C₂×C₃	—	3	4	no	—
140	C₂×C₃	—	6	4	no	—
141	C₂×C₃	—	18	2	no	—
142	C₂×C₃	C₂×C₈×C₃⋊D₃	1	1	yes	characteristic
143	D₃	—	6	8	no	—
144	C₂²	—	9	2	no	—
145	C₂²	C₈×#272	1	1	yes	characteristic
146	C₄	—	9	2	no	—
147	C₄	C₈×#272	1	1	yes	characteristic
148	C₄	C₂×C₄×#272	1	1	yes	characteristic
149	C₃	—	3	4	no	—
150	C₃	#10592606	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	18	2	no	—
153	C₂	C₂×C₈×#272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#11721776

C₂

yes

maximal, characteristic

H(3,𝔽₃)×M₃₂

C₂

yes

maximal, characteristic, complemented

C₂×#88386

C₂

yes

maximal, characteristic, complemented

#11720511

C₂

yes

maximal, complemented

C₁₆×#272

C₂

yes

maximal, complemented

C₃×#4907

—

maximal

#88740

C₄

yes

characteristic

C₂×C₈×H(3,𝔽₃)

C₂²

yes

characteristic

#88538

C₄

yes

characteristic

#88527

C₄

yes

—

C₈×#272

C₂²

yes

—

C₁₆×H(3,𝔽₃)

C₂²

yes

—

#88386

C₂²

yes

characteristic

#88386

C₂²

yes

characteristic

C₃×#901

—

C₃²×M₃₂

D₃

yes

complemented

C₂×C₃×C₃⋊C₁₆

—

C₃×#798

—

C₃×D₃×C₁₆

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

#6235

C₈

yes

—

C₄⋊#272

C₈

yes

characteristic

C₄×#272

C₂×C₄

yes

—

#6229

C₈

yes

characteristic

C₈×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₈×H(3,𝔽₃)

C₂³

yes

characteristic

#6180

C₂×C₄

yes

—

C₃×#504

—

#4907

—

C₃×#235

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃×#208

—

C₃×ΓL(1,𝔽₂₅)

—

C₃×D₃×C₈

—

C₃²×C₁₆

C₂×D₃

yes

—

C₃×C₃⋊C₁₆

—

C₃×C₃⋊C₁₆

—

C₂²×H(3,𝔽₃)

C₂×C₈

yes

characteristic

C₂×#272

C₂×C₈

yes

—

C₄×H(3,𝔽₃)

C₂×C₈

yes

characteristic

C₄×H(3,𝔽₃)

C₂²×C₄

yes

characteristic

#1084

C₂×C₈

yes

—

C₃×#131

—

#901

—

C₃×M₃₂

—

C₃×M₃₂

—

C₃×M₃₂

C₃⋊D₃

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

—

C₂×C₃⋊C₁₆

—

#798

—

D₃×C₁₆

—

C₂×C₃²×C₄

C₄×D₃

yes

—

C₃×#67

—

C₃×D₁₂

—

C₃×C₄×D₃

—

C₃×Dic₆

—

C₃²×C₈

C₄×D₃

yes

—

C₃²×C₈

C₂²×D₃

yes

—

C₃×C₃⋊C₈

—

#504

—

Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂²×C₈

yes

characteristic

#272

—

C₃×#41

—

#235

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

#208

—

ΓL(1,𝔽₂₅)

—

D₃×C₈

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

C₂×C₃⋊D₃

yes

—

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

D₃×C₈

yes

—

C₂×C₃×D₃

—

C₃²×C₄

D₃×C₈

yes

—

C₃²×C₄

C₂×C₄×D₃

yes

—

C₃×Dic₃

—

#131

—

M₃₂

—

M₃₂

#272

yes

characteristic, complemented

C₂×C₁₆

—

C₂×C₁₆

—

H(3,𝔽₃)

#504

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×C₃⋊D₃

yes

characteristic

#67

—

D₁₂

—

100

C₄×D₃

—

101

Dic₆

—

102

C₃×C₈

—

103

C₃×C₈

—

104

C₃×C₈

—

105

C₃×C₈

C₄×C₃⋊D₃

yes

characteristic

106

C₃×C₈

C₂²×C₃⋊D₃

yes

characteristic

107

C₃⋊C₈

—

108

C₂×C₃²

—

109

C₂×C₃²

C₂×D₃×C₈

yes

—

110

C₃×D₃

—

111

#41

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

C₂×#272

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

C₂×#272

yes

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

C₈×C₃⋊D₃

yes

characteristic

122

C₂×D₃

—

123

C₃×C₄

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

C₈×C₃⋊D₃

yes

characteristic

127

C₃×C₄

C₂×C₄×C₃⋊D₃

yes

characteristic

128

Dic₃

—

129

C₃²

#4907

yes

complemented

130

Q₈

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

C₄×#272

yes

characteristic

135

C₈

—

136

C₈

C₄×#272

yes

characteristic

137

C₈

C₂²×#272

yes

characteristic

138

C₂×C₃

—

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

C₂×C₈×C₃⋊D₃

yes

characteristic

143

D₃

—

144

C₂²

—

145

C₂²

C₈×#272

yes

characteristic

146

C₄

—

147

C₄

C₈×#272

yes

characteristic

148

C₄

C₂×C₄×#272

yes

characteristic

149

C₃

—

150

C₃

#10592606

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

C₂×C₈×#272

yes

characteristic

	subgroup	G/H
center, Frattini	C₃×C₈	C₂²×C₃⋊D₃
commutator	C₂×H(3,𝔽₃)	C₂²×C₈
Fitting	H(3,𝔽₃)×M₃₂	C₂
socle	C₂×C₃	C₂×C₈×C₃⋊D₃
prime radical	#88740	C₄
cosocle	C₈×H(3,𝔽₃)	C₂³
hypercenter	C₃×M₃₂	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#504
LCS	G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₃×C₈ ◁ C₃×M₃₂

subgroup

G/H

center, Frattini

C₃×C₈

C₂²×C₃⋊D₃

commutator

C₂×H(3,𝔽₃)

C₂²×C₈

Fitting

H(3,𝔽₃)×M₃₂

C₂

socle

C₂×C₃

C₂×C₈×C₃⋊D₃

prime radical

#88740

C₄

cosocle

C₈×H(3,𝔽₃)

C₂³

hypercenter

C₃×M₃₂

C₃⋊D₃

G ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₃×C₈ ◁ C₃×M₃₂

N	H ≅ G/N	type
2. H(3,𝔽₃)×M₃₂	C₂	semidirect
3. C₂×#88386	C₂	semidirect
4. #11720511	C₂	semidirect
5. C₁₆×#272	C₂	semidirect
16. C₃²×M₃₂	D₃	semidirect
88. M₃₂	#272	semidirect
91. H(3,𝔽₃)	#504	semidirect
129. C₃²	#4907	semidirect

H ≅ G/N

type

2. H(3,𝔽₃)×M₃₂

C₂

semidirect

3. C₂×#88386

C₂

semidirect

4. #11720511

C₂

semidirect

5. C₁₆×#272

C₂

semidirect

16. C₃²×M₃₂

D₃

semidirect

88. M₃₂

#272

semidirect

91. H(3,𝔽₃)

#504

semidirect

129. C₃²

#4907

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#11721776	2
H(3,𝔽₃)×M₃₂	2
C₂×#88386	2
#11720511	2
C₁₆×#272	2
C₃×#4907	3

subgroup

index

H(3,𝔽₃)×M₃₂

C₁₆×#272

Subgroups — #13144201

Conjugacy classes of subgroups

Complements

Maximal subgroups